I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.

I want to find a program in the case that it exists (does it?), or to program it. Here is my question: Can you give me a suggestion to program it? In particular, which software would you recommend me to use. I am also interested in any source of pictures of Markov partitions for toral automorphism in $\mathbb{T}^2$ (I would like to see other examples and their iterations rather than the classical by Weiss and Adler).

I want my program to do this:


i. A toral automorphism $T:\mathbb{T}^2\doteq\mathbb{R}^2/\mathbb{Z}^2\to \mathbb{T}^2$, i.e. a map that has linear lifting $L:\mathbb{R}^2\to \mathbb{R}^2$ without eigenvalues of modulus 1.

ii. A Markov partition $\alpha$. I want to see the image of $\mathbb{T}^2$ colored by $\alpha$.

iii. A positive integer $n$.


Plot of $T^n(\alpha)$. I want to see the image of $T^n(\mathbb{T}^2)$ when $\mathbb{T}^2$ was colored by $\alpha$.

Another question that I have. In the case that I had this program (that it assumes that we are able to find the Markov partition $\alpha$). Do you think that it will help to find Markov partitions (if instead of having a Markov partition as an input I put any topological partition)?


A practical way to solve my question is using SAGE (however, I think the code is not suitable on this website). I got easily a nice picture for a Markov partition for the toral automorphism that lifts to the linear map on $\mathbb{R}^2$ with matrix $M=(1,1,1,0).$

enter image description here

  • $\begingroup$ That's a nice picture. $\endgroup$ – Lee Mosher Mar 10 '14 at 16:16
  • $\begingroup$ Posting code here is perhaps not appropriate. But you should post it someplace, and make it easy to find! $\endgroup$ – Sam Nead Mar 10 '14 at 17:23

To address your second question, there is a simple construction of a Markov partition which inputs not "any topological partition" but which simply inputs the matrix $M \in SL_2(Z)$ that represents $T$. The algorithm will have to compute the eigenvectors and eigenvalues for $M$. Using that data the algorithm will compute the Markov partition which will consists of a certain pair of rectangles on $T$ with sides parallel to the eigenvectors. It is exactly like the classical Adler/Weiss examples.

  • 1
    $\begingroup$ Thanks. If I understand well, what I can do is to cover the torus with rectangles with sides parallels to the eigenvalues, and then to adjust the sizes to make it Markov. $\endgroup$ – user39115 Feb 26 '14 at 15:44
  • $\begingroup$ Yes, that's right. Two rectangles will do in this situation, and they can be constructed to have disjoint interiors. $\endgroup$ – Lee Mosher Feb 26 '14 at 16:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.